Do scientists know what is happening inside artificial neural networks?
There are many approaches that aim to make a trained neural network more interpretable and less like a "black box", specifically convolutional neural networks that you've mentioned.

Visualizing the activations and layer weights

Activations visualization is the first obvious and straight-forward one. For ReLU networks, the activations usually start out looking relatively blobby and dense, but as the training progresses the activations usually become more sparse (most values are zero) and localized. This sometimes shows what exactly a particular layer is focused on when it sees an image.

Another great work on activations that I'd like to mention is deepvis that shows reaction of every neuron at each layer, including pooling and normalization layers. Here's how they describe it:


  In short, we’ve gathered a few different methods that allow you to
  “triangulate” what feature a neuron has learned, which can help you
  better understand how DNNs work.


The second common strategy is to visualize the weights (filters). These are usually most interpretable on the first CONV layer which is looking directly at the raw pixel data, but it is possible to also show the filter weights deeper in the network. For example, the first layer usually learns gabor-like filters that basically detect edges and blobs.



Occlusion experiments

Here's the idea. Suppose that a ConvNet classifies an image as a dog. How can we be certain that it’s actually picking up on the dog in the image as opposed to some contextual cues from the background or some other miscellaneous object? 

One way of investigating which part of the image some classification prediction is coming from is by plotting the probability of the class of interest (e.g. dog class) as a function of the position of an occluder object. 
If we iterate over regions of the image, replace it with all zeros and check the classification result, we can build a 2-dimensional heat map of what's most important for the network on a particular image. This approach has been used in Matthew Zeiler’s Visualizing and Understanding Convolutional Networks (that you refer to in your question):



Deconvolution

Another approach is to synthesize an image that causes a particular neuron to fire, basically what the neuron is looking for. The idea is to compute the gradient with respect to the image, instead of the usual gradient with respect to the weights. So you pick a layer, set the gradient there to be all zero except for one for one neuron and backprop to the image. 

Deconv actually does something called guided backpropagation to make a nicer looking image, but it's just a detail.

Similar approaches to other neural networks

Highly recommend this post by Andrej Karpathy, in which he plays a lot with Recurrent Neural Networks (RNN). In the end, he applies a similar technique to see what the neurons actually learn:


  The neuron highlighted in this image seems to get very excited about
  URLs and turns off outside of the URLs. The LSTM is likely using this
  neuron to remember if it is inside a URL or not.


Conclusion

I've mentioned only a small fraction of results in this area of research. It's pretty active and new methods that shed light to the neural network inner workings appear each year.

To answer your question, there's always something that scientists don't know yet, but in many cases they have a good picture (literary) of what's going on inside and can answer many particular questions. 

To me the quote from your question simply highlights the importance of research of not only accuracy improvement, but the inner structure of the network as well. As Matt Zieler tells in this talk, sometimes a good visualization can lead, in turn, to better accuracy.